Question

What is the greatest common factor of `51x^3y^2 - 27xy + 69y`?

Answer 1

3y

Explanation:

I did this in two steps. I first looked at the numeric coefficients to determine if there was a common factor for the polynomial:

51 -27 69

51 is divisible by 3 and 17
27 is divisible by 3 and 9, and 9 is `3^2`, meaning `27 = 3^3`
69 is divisible by 3 and 23

since the shared factor among the three coefficients is 3, we can pull that out of the whole equation as a common factor:

`3(17x^3y^2-9xy+23y)`

Next, we can see if there are non-numeric coefficients (x and y in this case) that are used in all 3 terms. x is used twice, but y is found in all three terms. This means we can pull y out of the equation. You do this by dividing all 3 terms by y and putting a y outside the parentheses:

`3y(17x^3y-9x+23)`

The greatest common factor is the value outside of the parentheses in the above equation, meaing your answer is `color(red)(3y)`

Answer 2

`GCF(51x^3y^2,-27xy,69y)=color(red)(3y)`

Explanation:

Find the GCF of the constants and the composite variables separately:

`51=color(blue)3xx17`
`27=color(blue)3xx9`
`69=color(blue)3xx23`
`color(white)("XXX")`...by inspection `17,9, and 23` have no common factors `>1`

`x^3y^2=color(magenta)yxx x^3y`
`xy=color(magenta)y xx x`
`y=color(magenta)y`

Combining the factors: `color(blue)3color(magenta)y`